The T2 connection in Dublin is passport only If arriving into T2 IF

You arrive via a 400 gate (or bus gate),
OR
You arrive before the Emirates flight, having arrived via 300 gate (but routed to T2) if you arrive afterwards full security screening is required, only flights from EEA/UK/US/CA airports are considered exempt from screening.

T1 as its completely mixed you will pass through security and that assumes T1 connections are open, once back airside all gates are accessible from either terminal.

So

EI/AA to BA is easy
BA to EI/AA is messy